Feedback Filter Research Articles

A Joint Active Damping Strategy Based on LCL-Type Grid-Connected Inverters for Grid Current Feedback and PCC Voltage Unit Feedforward.

The negative high-pass filter feedback of the grid current (NFGCF) can offer active damping for the LCL-type grid-connected inverter. Due to the control delay in digital control systems, this damping can cause the system to exhibit non-minimum phase behavior within specific frequency ranges. This study proposes a joint active damping approach that combines grid current feedback and the point of common coupling (PCC) voltage unit feedforward. The proposed method introduces a dynamic damping region that varies with grid impedance. By developing suitable damping loop control parameters, this region can span the entire frequency range, even exceeding the Nyquist frequency fs/2. The research results demonstrate that the proposed approach enhances robustness against variations in grid impedance and eliminates non-minimum phase behavior. Simulation and experimental outcomes validate the effectiveness of this joint active damping method.

Multimodal information bottleneck for deep reinforcement learning with multiple sensors

Reinforcement learning has achieved promising results on robotic control tasks but struggles to leverage information effectively from multiple sensory modalities that differ in many characteristics. Recent works construct auxiliary losses based on reconstruction or mutual information to extract joint representations from multiple sensory inputs to improve the sample efficiency and performance of reinforcement learning algorithms. However, the representations learned by these methods could capture information irrelevant to learning a policy and may degrade the performance. We argue that compressing information in the learned joint representations about raw multimodal observations is helpful, and propose a multimodal information bottleneck model to learn task-relevant joint representations from egocentric images and proprioception. Our model compresses and retains the predictive information in multimodal observations for learning a compressed joint representation, which fuses complementary information from visual and proprioceptive feedback and meanwhile filters out task-irrelevant information in raw multimodal observations. We propose to minimize the upper bound of our multimodal information bottleneck objective for computationally tractable optimization. Experimental evaluations on several challenging locomotion tasks with egocentric images and proprioception show that our method achieves better sample efficiency and zero-shot robustness to unseen white noise than leading baselines. We also empirically demonstrate that leveraging information from egocentric images and proprioception is more helpful for learning policies on locomotion tasks than solely using one single modality.

Spatial diversity processing mechanism based on the distributed underwater acoustic communication system.

To address the problem of unreliable single-link underwater acoustic communication caused by large signal delays and strong multipath effects in shallow water environments, this paper proposes a distributed underwater acoustic diversity communication system (DUA-DCS). DUA-DCS employs a maneuverable distributed cross-medium buoy network to form multiple distributed, non-coherent, and parallel communication links. In the uplink, a receiving diversity processing mechanism of joint decision feedback equalizer embedded phase-locked loop and maximum signal-to-interference ratio combining (DFE-PLL-MSIRC) is proposed to achieve waveform-level diversity combining of underwater signals. A phase-locked loop module is embedded in each branch of the decision feedback equalizer to eliminate the residual frequency and phase errors after Doppler compensation. Meanwhile, the combining coefficients are determined based on the maximum signal-to-interference ratio criterion, taking into account the residual inter-symbol interference after equalization, resulting in efficient and accurate computation. Additionally, the combined decision values are fed back to the feedback filters in each branch to ensure more accurate feedback output. Simulation and lake experiment results demonstrate that, compared to the single-link communication system, DFE-PLL-MSIRC can achieve a diversity gain of more than 5.2 dB and obtain about 3 dB more diversity gain than the comparison algorithm. And the BER of DFE-PLL-MSIRC can be reduced by at least one order of magnitude, which is lower by at least 0.6 order of magnitude compared to the comparison algorithm. In the downlink, a transmitting diversity processing mechanism of complex orthogonal space-time block coding (COSTBC) is proposed. By utilizing a newly designed generalized complex orthogonal transmission matrix, complete transmission diversity can be achieved at the coding rate of 3/4. Compared to the single-link communication system, the system can achieve a diversity gain of more than 6 dB.

Virtual Space-Time DiversityTurbo Equalization Using Cluster Sparse Proportional Recursive Least Squares Algorithm for Underwater Acoustic Communications

The oceanic positioning, navigation and timing (PNT) network requires high-quality underwater acoustic message transmission. Turbo equalization technology has exhibited superior performance for underwater acoustic (UWA) communications compared with conventional channel equalizers. To overcome the performance reduction caused by severe doubly selective UWA channels, the virtual space-time diversity soft direct-adaptation turbo equalization is proposed for UWA communications. The proposed scheme improves the ability of the typical turbo equalizer to deal with both Doppler and multipath effects for time varying channels. We utilize a fractionally spaced soft interference cancellation equalizer (FS-SE) instead of a hard decision to constitute the soft-input soft-output (SISO) equalizer. Combined with another virtual time-reversal mirror equalizer component, we can obtain virtual space and time diversity with only a single receiving transducer and mitigate the error propagation phenomenon of the feedback filter. To satisfy the sparse UWA channel, the ℓp,q-PRLS algorithm is applied to adaptive updates for FS-SE. In the proposed scheme, an adjustable interpolator and digital phase-locked loop are embedded into the equalizer to overcome the residual Doppler frequency shift and recover the timing distortion. Results of simulations and field lake trial show that the proposed scheme achieves better performance than existing ones under the same equalizer order.

Photonic Millimeter Wave Generation and Stabilization in Optically Injected Discrete-Mode Semiconductor Lasers Subject to Photonic Filter Feedback

Photonic millimeter wave signal generation and stabilization based on nonlinear dynamics of optically injected discrete-mode semiconductor lasers with photonic filter feedback are experimentally and numerically studied. The photonic filter is constructed by jointing two ports of a 2×2 optical coupler to form a ring cavity recirculation and is modelled as an infinite impulse response filter. The results show that >30GHz photonic millimeter wave signals can be obtained after optical to electrical conversion of the period-one oscillation output of the optically injected discrete-mode semiconductor laser. More importantly, the linewidth, side peak suppression ratio, as well as the stability of the generated millimeter wave, can be optimized using the photonic filter feedback. A fair comparison of the photonic filter feedback scheme and the single/double optical feedback schemes in terms of optimization performance has been made. The corresponding results demonstrate that the photonic filter feedback scheme has obvious superiority in millimeter wave side peak suppression and stability. The effect of the coupling coefficient as well as the phase variables in the ring cavity has also been discussed in the simulation work and the results qualitatively agree with the experimental observations.

Globally optimal distributed and sequential state fusion filters for multi-sensor systems with correlated noises

The problem concerning the multi-sensor fusion filtering for systems with correlated system and measurement noises is studied in this paper. Each sensor produces a local predictor by using its own measurements and then sends it to a fusion center. In the fusion center, by applying the projection theory and difference technology, the globally optimal distributed state fusion filter without feedback and sequential state fusion filter with recursive estimators of noises are presented in the sense of linear minimum variance by using all received local predictors, respectively. Under the condition that local prediction gain matrices are of full column rank, the proposed distributed and sequential state fusion filters can achieve the same estimation accuracy as the centralized fusion filter, i.e., they both have global optimality. The optimality is strictly proved. The corresponding steady-state fusion filters are also explored. A sufficient condition for the convergence of the proposed fusion filters is given. A target tracking example verifies the effectiveness of the proposed algorithms.

First-Order Linear Active Disturbance Rejection Control for Turbofan Engines

Proportional-integral (PI) control is widely used in turbofan-engine control, while first-order linear active disturbance rejection control (FOLADRC) is a possible approach to update it. This paper investigates FOLADRC. In methodology, it proposes a new block diagram of FOLADRC, which shows that FOLADRC can be viewed as a PI controller, a low-pass feedback filter, and a pre-filter. The low-pass filter helps to reject high-frequency measurement noise, while the pre-filter can attenuate overshoot in step response. In simulation, 14 published linearized model matrices of NASA’s CMAPSS-1 90k engine model are used to verify the above theory. Simulations show one FOLADRC controller can be simultaneously used for the 14 linear models and guarantee that all the 14 low-pressure turbine speed control loops have enough phase margin and no overshoot. Thus, replacing several PI controllers with one FOLADRC controller is possible, and FOLADRC can be used to simplify the control system design of turbofan engines.

Design of Synchronization Tracking Adaptive Control for Bilateral Teleoperation System with Time-Varying Delays.

The performances of position synchronization and force interaction of the teleoperation system provide a safe and efficient way for operators to perform tasks in remote, hazardous environments. In practice, however, communication delays and dynamic uncertainties can impair the performance of position synchronization controls. Under the above factors, it is necessary to study and design appropriate bilateral control methods to achieve stable and effective position synchronization control. In this paper, a new adaptive control architecture based on velocity feedback filter and radial basis function neural network is proposed. In the proposed control scheme, only the position signal is transmitted during the communication process, and the speed feedback filter and compensation method are designed and adopted to avoid the use of acceleration signals. In addition, a new auxiliary variable with a tracking error integral term is used to reduce the steady-state error of position tracking under nonzero external environmental forces. Using the Lyapunov–Krasovskii method, the stability of closed-loop remote operating systems is demonstrated. In the simulation and experiment sections, the algorithm was verified separately and compared with other algorithms. The results of a master–slave robot system verify the tracking performance of our proposed control scheme.

Suppression of Wind Generator Speed Vibration Based on the Internal Model Control with Three Degrees of Freedom

When the wind power system is working in the maximum wind energy tracking area, or when its mechanical drivetrain system vibrates and the vibration frequency is relatively high, the system cannot suppress the speed vibration of the generator in the drivetrain system by adjusting the pitch angle. In this paper, a generator speed control system based on the three degrees of freedom internal model control (3-DOF-IMC) is established to control the generator speed. Thus, a scheme of using the feedback filter in the 3-DOF-IMC to suppress generator speed vibration caused by drivetrain shaft elasticity and gear clearance in the drivetrain system is proposed. Firstly, the vibration problem and waveform of the two-mass wind power drivetrain systems are discussed, and the generator’s vector control and speed control systems are analyzed. Secondly, the principle of the 3-DOF-IMC is described, and the influence of the three controllers on the speed tracking performance and anti-interference performance of the generator is discussed. The suppression ability of the feedback filter for different forms of disturbance signals is emphasized. Finally, the feedback filter controls the generator speed and eliminates the influence of drivetrain vibration on the generator speed. To verify the superiority of the proposed method, the vibration suppression effect, tracking performance and anti-interference performance of the proposed method are compared with the engineering design method (EDM) and conventional IMC (1-DOF-IMC) method. When the parameter α/β = 0.66, the generator speed amplitude overshoot of the proposed method is the same as the EDM. When α/β = 1, it is only 4% of the amplitude overshoot of the EDM. In addition, the tracking performance and anti-interference performance of the proposed method can be adjusted independently, and it is better than the EDM and the 1-DOF-IMC method.

Firefly Algorithm based LCL filtered grid-tied STATCOM design for reactive power compensation in SCIG based Micro-grid

Microgrids are facing several operational and control issues while integrating with the grid. To deal with it, STATCOM, as one of the emerging power converter circuits, is connected with such microgrids. STATCOM with microgrid introduces current harmonic, inherent resonance, and active power losses related to high switching frequency. Filter design can help attenuate these effects and maintain the predefined standards as in IEEE 519-1992 and IEEE P1547.2-2003. There are also some more points of concern about filter design. Inappropriate filter types and parameters may cause worse filtering, reactive power surplus production, and low power factor. Therefore, this paper suggests that the reactive power compensation capability, as a local area problem, must be attended through the proper designing of LCL filtered grid-tied STATCOM. The parameters are estimated through transient analysis, power quality, and power balance studies for proposed Micro-grid and the results obtained are compared using nature-inspired algorithms such as GA, PSO, and FA over conventional mathematical formulation. The main contributions of this work are; (i) study of system with the inclusion of voltage and frequency-dependent load, (ii) modified STATCOM model clubbing reactive power control feedback and LCL filter with damping resistance, (iii) real and reactive power tracking system using dynamic compensator capabilities, and (iv) suppression of total harmonic distortions along with real and reactive power tracking using advanced tuning for filter parameters with GA, PSO, and FA based algorithms.

Chaotic LIF Oscillator With Variable Resistance Feedback and Nonlinear Rate Coding

This brief presents a model of chaotic oscillator based on leaky integrate-and-fire (LIF) neuron with a switching element, which has an S-shaped current-voltage characteristic (S-IVC), and with nonlinear firing rate coding by control resistor. The circuit contains variable resistance feedback of LIF neuron, which includes a second order filter and two gain modules. Simulation results show that active RC-circuit can be used as a feedback filter and a bipolar transistor can be used as a control resistor. A mathematical model of the chaotic oscillator has been built. The transition to the dynamic chaos occurs according to the scenario of a period-doubling bifurcation cascades. The circuit provides theoretical perspectives for the creating devices of neuromorphic computing based on chaotic dynamics.

Coupling Matrix-Based Co-Design of Filter-Oscillators

This letter presents a methodology of designing integrated filter-oscillators using the coupling matrix technique. The <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N + 4$ </tex-math></inline-formula> coupling matrix is developed for the first time to describe the filter-oscillator topology. Accordingly, the corresponding group delay, loop gain, loop phase, and complex quality factor of the feedback filter can be calculated using the matrix. This helps rapidly predict oscillating frequencies at group delay or complex quality factor peak frequencies to achieve lower phase noise. Moreover, since the transistor and the transmission line (TL) entries are incorporated in the coupling matrix, the physical geometries can be directly determined, resulting in improved design efficiency. Two third-order filter-oscillators are implemented at 2.91 GHz (group delay peak) and 3.15 GHz (complex quality factor peak) frequencies, respectively. The developed filter-oscillators are measured with −121 and −145 dBc/Hz phase noise at 1-MHz offset frequency, validating this coupling matrix-based co-design approach.

Design of 7 × 2.5 Gbps decision feedback equalization scheme for mode division multiplexing over free-space optics under diverse atmospheric turbulence

Free-space optics (FSO) systems use the atmosphere as a medium to propagate data in the form of an optical signal. Using mode division multiplexing (MDM) over FSO systems increases the capacity of the FSO link. However, atmospheric turbulence is one of the main problems that seriously limit the data transmission rate in FSO systems. This causes signal distortion during transmission that leads to data loss at the receiver side totally or partially. To address this problem, a decision feedback equalization (DFE) scheme with minimum mean square error (MMSE) in MDM for FSO systems was developed in OptSim 5.2 software to mitigate the adverse effects of atmospheric turbulence. The MMSE algorithm was used to optimize both the feed-forward and feedback filter coefficients of the DFE. The proposed DFE scheme in MDM for the FSO system comprises seven parallel 2.5 Gbps MDM channels using Hermite–Gaussian modes at the transmitter to transmit 17.5 Gbps and seven DFE schemes at the receiver to mitigate the effects of atmospheric turbulence. The simulation results demonstrate that the DFE scheme can transmit 17.5 Gbps over 40 m, 800 m, 1400 m, and 2 km under medium fog, medium rain, medium haze, and clear weather, respectively. We show that the DFE scheme improves the MDM over FSO system immunity to distortion in medium fog, medium rain, medium haze, and clear weather while maintaining high throughput and desired low bit error rate.

The Issue of Proxies and Choice Architectures. Why EU Law Matters for Recommender Systems.

Recommendations are meant to increase sales or ad revenue, as these are the first priority of those who pay for them. As recommender systems match their recommendations with inferred preferences, we should not be surprised if the algorithm optimizes for lucrative preferences and thus co-produces the preferences they mine. This relates to the well-known problems of feedback loops, filter bubbles, and echo chambers. In this article, I discuss the implications of the fact that computing systems necessarily work with proxies when inferring recommendations and raise a number of questions about whether recommender systems actually do what they are claimed to do, while also analysing the often-perverse economic incentive structures that have a major impact on relevant design decisions. Finally, I will explain how the choice architectures for data controllers and providers of AI systems as foreseen in the EU's General Data Protection Regulation (GDPR), the proposed EU Digital Services Act (DSA) and the proposed EU AI Act will help to break through various vicious circles, by constraining how people may be targeted (GDPR, DSA) and by requiring documented evidence of the robustness, resilience, reliability, and the responsible design and deployment of high-risk recommender systems (AI Act).

Delay-Doppler domain decision feedback turbo equalization for OTFS modulation

Emerging evidence indicates that orthogonal time–frequency space (OTFS) modulation is a potential candidate modulation scheme for high mobility wireless communications. However, OTFS may experience significant inter-symbol interference (ISI) and inter-Doppler interference (IDI) in the receiver. In this paper, we propose a soft decision feedback turbo equalization for OTFS transmission over delay-Doppler channels to jointly combat both interferences. A novel block decision feedback equalization (BDFE) algorithm is constructed using the band feature of the channels in the delay-Doppler domain. The feedforward and feedback filters are designed by the delay-time channels coefficients. According to the designed filter, an equivalent system model is employed to allow turbo equalization. The posterior probability is established using the soft prior information and feedback filter, and then fed back to the channel decoder as external information. Both theoretical analysis and simulation results demonstrate the effectiveness of the proposed scheme in improving the bit error rate (BER) performance and combat various interference. Numerical simulations are finally provided to justify the validity of the proposed scheme in improving the bit error rate (BER) performance and combating various interference.

HM-Net: A Regression Network for Object Center Detection and Tracking on Wide Area Motion Imagery

Wide Area Motion Imagery (WAMI) yields high resolution images with a large number of extremely small objects. Target objects have large spatial displacements throughout consecutive frames. This nature of WAMI images makes object tracking and detection challenging. In this paper, we present our deep neural network-based combined object detection and tracking model, namely, Heat Map Network (HM-Net). HM-Net is significantly faster than state-of-the-art frame differencing and background subtraction-based methods, without compromising detection and tracking performances. HM-Net follows object center-based joint detection and tracking paradigm. Simple heat map-based predictions support unlimited number of simultaneous detections. The proposed method uses two consecutive frames and the object detection heat map obtained from the previous frame as input, which helps HM-Net monitor spatio-temporal changes between frames and keeps track of previously predicted objects. Although reuse of prior object detection heat map acts as a vital feedback-based memory element, it can lead to unintended surge of false positive detections. To increase robustness of the method against false positives and to eliminate low confidence detections, HM-Net employs novel feedback filters and advanced data augmentations. HM-Net outperforms state-of-the-art WAMI moving object detection and tracking methods on WPAFB dataset with its 96.2% F1 and 94.4% mAP detection scores, while achieving a 61.8% mAP tracking score on the same dataset.

Fast physical random bit generation of wideband flat chaos signal based on filter feedback

&lt;sec&gt;Chaotic lasers feature wide spectrum and noise-like features, and extensively used in various fields, such as secure communications and random bit generation (RBG). Since the physical RBG using optical chaos was demonstrated first by Uchida et al., the optical chaos has been widely investigated in terms of chaos bandwidth and flatness, which determines the rate and randomness of RBG. Owing to the natural stability of semiconductor lasers, external perturbation is required to generate chaotic signals, such as optical injection, current modulation, and optical feedback. Among them, a semiconductor laser with optical feedback has attracted wide attention because of its simple structure and rich dynamic behaviors. Nonetheless, this configuration suffers the influence of the relaxation oscillation, which results in a limited bandwidth (a few GHz) and an uneven power spectrum. To obtain broad-spectrum chaotic signals, considerable efforts have been made in recent years. However, these solutions are associated with complex structures that require delicate manipulation because multiple parameters should be matched, so the cost of some of these schemes in terms of the system complexity can potentially outweigh the benefits.&lt;/sec&gt;&lt;sec&gt;In this work, we incorporate an optical filter and an amplifier into the feedback loop of a conventional optical feedback system to generate broadband chaotic signals. The effects of the filter detuning frequency and feedback power on the bandwidth and flatness of the chaotic output are investigated experimentally. The experimental results demonstrate that by appropriately adjusting the feedback power and detuning frequency, both the low-frequency components and the high-frequency components of the chaotic output power spectrum can be increased, and the maximum chaotic bandwidth can reach 24.4 GHz with a flatness of 5.7 dB. This phenomenon is attributed to the physical process of beating between the filtered mode and the internal modes of the laser. Furthermore, the optimized chaotic output is processed by retaining the 4 least significant bits and implementing the delayed exclusive-OR (XOR) operation. Our scheme is capable of generating physical random number of the bit rate of 320 Gbit/s, and successfully passes the standard randomness test, i.e. the NIST test (NIST SP 800-22).&lt;/sec&gt;

Experimental Research on the Noise Characteristics of the Output Field of the Optical Filter Cavity

Precision measurement is an important direction of today's frontier scientific research. Using lasers to achieve high-precision target measurement has become an important way to improve measurement accuracy, which can be applied in various fields. However, for a certain application, the measurement accuracy will directly depend on the noise level of the laser source. Most applications require that the measurement frequency band is concentrated in the audio frequency band. In order to obtain a low-noise laser source with shot noise limited in the applied frequency band, active and/or passive noise reduction are the usual choice, i.e., active feedback control and filter cavity technique, and so on. Therefore, noise analysis and suppression techniques are the main concern of the precision measurement. The optical filter cavity acts as an optical low-pass filter, which can effectively suppress high-frequency noise beyond its linewidth. In this work, we found that the intensity noise of the output field of an optical filter cavity is higher than the noise floor of the laser. The main sources of noise are analyzed through experiments:(1) excess noise introduced by cavity length locking; (2) laser phase and pointing noises coupled to the intensity one by the cavity. To cancel the excess noise as much as possible, we optimize the feedback control loop by measuring the open-loop and closed-loop transfer functions of the MC, combined with the critical proportionality method. All the control loop are homemade, and the PID is designed with a FPGA board for expediently achieving a noise reduction up to 30 dB at the audio frequency. Then the control loop is optimized as the best condition without introducing the excess noise. Compared with the free-running laser, MC filters out the high-frequency noise, meanwhile converts the phase noise and pointing noise of input field into the intensity noise of the output field. Therefore, the power noise spectrum in the audio segment is still higher than that of the input optical field itself. In the future, an active control loop will be applied to suppress the noise power. The experimental results provide the basic means for applied research such as feedback control loop noise analysis, which will promote the development of precision measurement to higher measurement accuracy.

Experimental study on noise characteristics of audio frequency band in output field of optical filter cavity

Precision measurement is an important direction of today’s frontier scientific research. Using lasers to achieve high-precision target measurement has become an important way to improve measurement accuracy, which can be used in various fields. However, for a certain application, the measurement accuracy will directly depend on the noise level of the laser source. Most of applications require that the measurement frequency band is concentrated in the audio frequency band. In order to obtain a low-noise laser source with shot noise limited in the applied frequency band, active and/or passive noise reduction is usually an option, i.e. active feedback control or filter cavity technique, etc. Therefore, noise analysis and suppression techniques are the main concern of the precision measurement. The optical filter cavity acts as an optical low-pass filter, which can effectively suppress high-frequency noise beyond its linewidth. In this work, we find that the intensity noise of the output field of an optical filter cavity is higher than the noise floor of the laser. The main sources of noise are analyzed experimentally, showing that 1) excess noise is introduced by cavity length locking, and 2) laser phase and pointing noises are coupled to the intensity one by the cavity. To cancel the excess noise as much as possible, we optimize the feedback control loop by measuring the open-loop and closed-loop transfer functions of the mode cleaner (MC), combined with the critical proportionality method. All the control loops are homemade, and the proportional-integral-derivative (PID) is designed with a field programmable gate array board for expediently achieving a noise reduction up to 30 dB at the audio frequency. Then the control loop is optimized to the best condition without introducing the excess noise. Compared with the free-running laser, MC filters out the high-frequency noise, meanwhile converts the phase noise and pointing noise of input field into the intensity noise of the output field. Therefore, the power noise spectrum in the audio band is still higher than that of the input optical field itself. In the future, an active control loop will be used to suppress the noise power. The experimental results provide the basic means for application research such as feedback control loop noise analysis, which will promote the development of precision measurement toward higher measurement accuracy.

NOISE ANALYSIS AND OPTIMIZATION OF GENERAL OTA-C FILTERS

In the paper, a general approach to noise analysis in continuous-time OTA-C filters is presented. Recently an increasing interest in the design of continuous-time filters based on the transconductance capacitor (OTA-C) technique has been observed. The operational transconductance amplifiers (OTAs) offer a higher bandwidth than their voltage-mode counterparts, can be easily tuned electronically, and have a better suitability for operating in reduced supply environment. Due to this, high frequency integrated filters are mostly realized as the OTA-C ones. Based on a matrix description of a general OTC filter topology, a universal formulas for evaluating the noise in any OTC filter are derived. The presented formulas can be easily implemented and used in computer-aided analysis/optimization software. The accuracy of the proposed method is confirmed by comparison with SPICE simulation. The example of application for finding the minimum-noise 5th order multiple-loop feedback filters implementing Butterworth and Bessel transfer functions is given.